Griffith microcrack criterion

7.4.1 Griffith microcrack criterion

Most materials break at a stress well below the theoretical fracture stress, which is that stress, σ t , required to pull apart two adjoining layers of atoms. This stress varies with the distance between the atom planes and, as shown in Figure 7.29, may be approximated to a sine curve of wavelength such that

where u is the displacement from the equilibrium spacing b. From Hooke’s law σ = (Eu/b) and hence σ t = λE/2πb. Now in pulling apart the two atomic planes it is necessary to supply the surface energy γ and hence

so that the theoretical tensile strength is given by σ t = (Eγ/b).

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Stress

Figure 7.29 Model for estimating the theoretical fracture strength σ t .

Glass fibers and both metallic and non-metallic whiskers have strengths approaching this ideal value of about E/10, but bulk metals, even when tested under favorable conditions (e.g. at 4 K), can rarely withstand stresses above E/100. Griffith, in 1920, was the first to suggest that this discrepancy was due to the presence of small cracks which propagate through the crystal and cause fracture. Griffith’s theory deals with elastic cracks, where at the tip of the crack atomic bonds exist in every stage of elongation and fracture. As the crack grows, each of the bonds in its path take up the strain, and the work done in stretching and breaking these bonds becomes the surface energy γ of the fractured faces. If separation of the specimen between two atomic layers occurs in this way, the theoretical strength only needs to be exceeded one point at a time, and the applied stress will be much lower than the theoretical fracture stress. The work done in breaking the bonds has to be supplied by the applied force, or by the elastic energy of the system. Assuming for a crack of length 2c that an approximately circular region

of radius c is relieved of stress σ and hence strain energy (σ 2 / 2E)πc 2 by the presence of the crack, the condition when the elastic strain of energy balances the increase of surface energy is given by

(2cγ)

dc 2E dc

and leads to the well-known Griffith formula, %

for the smallest tensile stress σ able to propagate a crack of length 2c. The Griffith criterion therefore depends on the assumption that the crack will spread if the decrease in elastic strain energy resulting from an increase in 2c is greater than the increase in surface energy due to the increase in the surface area of the crack.

Griffith’s theory has been verified by experiments on glasses and polymers at low temperatures, where a simple process of fracture by the propagation of elastic cracks occurs. In such ‘weak’ brittle fractures there is little or no plastic deformation and γ is mainly the surface energy ( ≈1–10 J m −2 ) and

E. In crystalline solids, however, the cracks are not of the elastic type and a plastic zone exists around the crack tip, as shown in Figure 7.30. In such specimens, fracture cannot occur unless the applied tensile stress σ is magnified to the theoretical strength σ t . For an atom- ically sharp crack (where the radius of the root of the crack r is of the order of b) of length 2c it can be shown that the magnified stress σ m will be given by σ m =σ (c/r), which, if the crack is to propagate, must be equal to the theoretical fracture stress of the material at the end of the crack. It follows that substituting this value of σ t in equation (7.22) leads to the Griffith formula of equation (7.23).

the fracture strength σ f ≈ 10 −5

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Figure 7.30 Variation of stress from the tip of a crack and the extent of the plastic zone, radius r y .

Figure 7.30 shows the way the magnified stress drops off with distance from the tip of the crack. Clearly, at some distance r y the stress reaches the yield stress and plastic flow occurs. There is thus

a zone of plastic flow around the tip of radius r y . The larger the plastic zone, as in ductile metals, the more energy is absorbed in fracture. In ceramics this zone is usually small. In ‘strong’ fractures γ is greatly increased by the contribution of the plastic work around the crack tip, which increases the work required for crack propagation. The term γ must now be replaced by (γ +γ p ), where γ p is the plastic work term; generally (γ +γ p ) is replaced by G, the strain energy release rate, so that equation (7.23) becomes the Orowan–Irwin relationship:

σ = (EG/πc). (7.24) Here, G might be

∼10 4 Jm −2 and σ f ≈ 10 −2 –10 −3 E.

Worked example

Steel with a lattice friction stress of 6.9 × 10 8 Nm −2 , dislocation density of 10 16 m −2 , surface energy γ s =1Jm −2 contains microcracks 10 −4 m in length. Will the component yield or fracture in service?

Solution

Assuming direct stress is two times the shear stress, to yield, σ

8 9 √ y =σ i + 4μb ρ = 2 × 6.9 × 10 + 4 × 8 × 10 × 0.2 × 10 −9 × 10 16

8 8 = 13.8 × 10 8 + 6.4 × 10 = 20.2 × 10 Pa = 2.02 GPa. To fracture, σ

f = 2Eγ s /π a = 8 2 × 210 × 10 9 × 1/π × 10 −4 = 0.36 × 10 Pa. σ f ≪σ y , so fracture will take place before yield.

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Plane stress (100% shear lip)

Plane strain c 0% shear lip

K K 1c

Section thickness

Figure 7.31 Variation in the fracture toughness parameter with section thickness.